Is stress tensor a 3x3 or 3x3x3 tensor?

[Hassan2]

Hi all,

I have a fundamental question about ( mechanical) stress tensor. Stress tensor a 3x3 tensor whose 9 entries looks "scalars" but in figures, the stress is illustrated by nine "vectors". Does it mean the stress tensors is in fact a 3x3x3 tensor of scalars whose nonzero entries are ignored?

 

[AlephZero]

Stress and strain are definitely 3x3 tensors.

If you mean diagrams of a rectangular block with arrows attached to the faces, the arrows are showing the direction of the forces acting on the face, i.e. the components of $\sigma \cdot \bf{n}$ where $\bf{n}$ is the normal vector to the face.

If you meant some other diagram, can you post a link to it?

 

[Hassan2]

Yes those diagrams that you mentioned. So when they write σ₁₁ beside one of the arrows, the arrow is the force component in that defined direction andσ₁₁ tells us about "magnitude" of the force. However it seems to be the force per unit area, otherwise the stress must be multiplied by the surface.

Thanks.

 

[AlephZero]

So when they write σ₁₁ beside one of the arrows, the arrow is the force component in that defined direction andσ₁₁ tells us about "magnitude" of the force. However it seems to be the force per unit area, otherwise the stress must be multiplied by the surface.

Correct.

 

[PJC01]

I can confirm that the stress tensor is indeed a 3x3 tensor. This means that it has 9 entries, each of which is a scalar value representing the stress in a particular direction. The use of vectors in illustrations is simply a visual representation of these scalar values and does not change the fact that the stress tensor is a 3x3 tensor. The use of a 3x3x3 tensor for stress would not be mathematically accurate and would not accurately represent the physical properties of stress. Therefore, it is important to understand that the stress tensor is a 3x3 tensor, not a 3x3x3 tensor.